Appendix Discounting
Formulae and
Tables
The general discounting formula,
as
noted in the introductory
chapter,
is:
present value = net benefit0 + net benefit1 (D1) +
...
. . . net benefitn
(Dn)
where the net benefit terms are the result of adding together
negative and positive terms in the income stream and D1
•••
Dn
are discount factors:
11(1
+ rY
...
11(1
+
rt
for years 1 to n.
Software exists to handle present value calculations without the
analyst having to work through tables
of
discount factors, but
there are obvious intellectual benefits in understanding what
is
at
issue in any particular discounting problem. Consequently it
is
worthwhile showing how the basic formula may be manipulated to
perform simple calculations. Where the time stream contains a
constantly recurring annual net benefit term (or where either the
cost
or
benefit stream has a constant annual datum) there
is
a
useful formula which can be employed. The basic
PV
formula
is:
PV
=
NB
[11(1
+
rY
+
11(1
+
r)
2 +
...
11(1
+
rt]
(1)
where
NB
represents
NB
0,
NB
1
•••
NB
0, the annually recurring
net benefit term.
If
we multiply (1) by
(1
+
r)
we obtain:
PV
(1
+
r)
=
NB
[(1
+ r)/(1 +
r)
+
(1
+ r)/(1 +
r)
2
+
(1
+ r)/(1 +
r)
3 +
...
(1
+ r)/(1 +
r)n]
(2)
which may be simplified to:
PV
(1
+
r)
=
NB
[1
+
11(1
+
r)
+
11(1
+
r)
2
+
...
1/(1 + rt-1] (3)
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